수열의 귀납적 정의 연습문제 — Sundstrom, Mathematical Reasoning · 4.3절 · 2a번

Sundstrom, Mathematical Reasoning · 4.3절 · 2a번영어 원문

Assume that $f_1 ,f_2 , \ldots ,f_n , \ldots$ are the Fibonacci numbers. Prove each of the following:

For each $n \in \mathbb{N}$ , $f_{4n}$ is a multiple of 3.

정답·풀이 보기 (영어 원문)

Let $P( n )$ be, “ $f_{4n}$ is a multiple of 3.” Since $f_4 = 3$ , $P( 1 )$ is true. If $P( k )$ is true, then there exists an integer $m$ such that $f_{4k} = 3m$ . We now need to prove that $P(k+1)$ is true or that $f_{4(k+1)}$ is a multiple of 3. We use the following:

$\begin{aligned}f_{4 \left( k + 1 \right)} & = f_{4k + 4} \\ & = f_{4k+3} + f_{4k+2} \\ & = \left( f_{4k+2} + f_{4k+1} \right) + \left( f_{4k+1} + f_{4k} \right) \\ & = f_{4k+2} + 2 f_{4k+1} + f_{4k} \\ & = \left( f_{4k+1} + f_{4k} \right) + 2 f_{4k+1} + f_{4k} \\ & = 3f_{4k + 1} + 2f_{4k}\end{aligned}$

We now use the assumption that $f_{4k} = 3m$ and the last equation to obtain $f_{4(k+1)} = 3f_{4k + 1} + 2\cdot 3m$ and hence, $f_{4(k+1)} = 3 \left( f_{4k +1} + 2m \right)$ . Therefore, $f_{4(k+1)}$ is a multiple of 3 and this completes the proof of the inductive step.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 4.3절 · 2c번영어 원문

Assume that $f_1 ,f_2 , \ldots ,f_n , \ldots$ are the Fibonacci numbers. Prove each of the following:

For each $n \in \mathbb{N}$ with $n \geq 2$ , $f_1 + f_2 + \cdots + f_{n - 1} = f_{n + 1} - 1$ .

다 풀었으면 정답 확인

Let $P( n )$ be, “ $f_1 + f_2 + \cdots + f_{n - 1} = f_{n + 1} - 1$ .” Since $f_1 = f_3 - 1$ , $P ( 2 )$ is true. For $k \geq 2$ , if $P ( k )$ is true, then $f_1 + f_2 + \cdots + f_{k - 1} = f_{k + 1} - 1$ . Then

$\begin{aligned}\left( f_1 + f_2 + \cdots + f_{k - 1} \right) + f_k & = \left( f_{k + 1} - 1 \right) + f_k \\ & = \left( f_{k+1} + f_k \right) - 1 \\ & = f_{k+2} - 1\end{aligned}$

. This proves that if $P \left( k \right)$ is true, then $P \left( k + 1\right)$ is true.

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